\(\int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \sqrt {a+c x^4}} \, dx\) [121]

Optimal result

Integrand size = 35, antiderivative size = 1059 \[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \sqrt {a+c x^4}} \, dx =\text {Too large to display} \] Output:

-1/2*a^(1/2)*C*d*(c+a/x^4)*(e+d/x^2)*x^3/c/e/(a*e^2+c*d^2)^(1/2)/(1+a^(1/2 
)*(e+d/x^2)/(a*e^2+c*d^2)^(1/2))/(e*x^2+d)^(1/2)/(c*x^4+a)^(1/2)+1/2*C*(e* 
x^2+d)^(1/2)*(c*x^4+a)^(1/2)/c/e/x-1/4*(-2*B*e+C*d)*(c+a/x^4)^(1/2)*(e+d/x 
^2)^(1/2)*x^3*arctanh(c^(1/2)*(e+d/x^2)^(1/2)/e^(1/2)/(c+a/x^4)^(1/2))/c^( 
1/2)/e^(3/2)/(e*x^2+d)^(1/2)/(c*x^4+a)^(1/2)+1/2*a^(1/4)*C*(a*e^2+c*d^2)^( 
3/4)*(1+a^(1/2)*(e+d/x^2)/(a*e^2+c*d^2)^(1/2))*(d^2*(c+a/x^4)/(a*e^2+c*d^2 
)/(1+a^(1/2)*(e+d/x^2)/(a*e^2+c*d^2)^(1/2))^2)^(1/2)*(e+d/x^2)^(1/2)*x^3*E 
llipticE(sin(2*arctan(a^(1/4)*(e+d/x^2)^(1/2)/(a*e^2+c*d^2)^(1/4))),1/2*(2 
+2*a^(1/2)/(a*e^2+c*d^2)^(1/2)*e)^(1/2))/c/d/e/(e*x^2+d)^(1/2)/(c*x^4+a)^( 
1/2)-1/2*(a^(1/2)*c*d^2*(-B*e+C*d)+a^(3/2)*e^2*(-B*e+C*d)+e*(A*c*d+B*a*e-C 
*a*d)*(a*e^2+c*d^2)^(1/2))*(1+a^(1/2)*(e+d/x^2)/(a*e^2+c*d^2)^(1/2))*((c+a 
/x^4)/(c+a*e^2/d^2)/(1+a^(1/2)*(e+d/x^2)/(a*e^2+c*d^2)^(1/2))^2)^(1/2)*(e+ 
d/x^2)^(1/2)*x^3*InverseJacobiAM(2*arctan(a^(1/4)*(e+d/x^2)^(1/2)/(a*e^2+c 
*d^2)^(1/4)),1/2*(2+2*a^(1/2)/(a*e^2+c*d^2)^(1/2)*e)^(1/2))/a^(1/4)/c/d^2/ 
e/(a*e^2+c*d^2)^(1/4)/(e*x^2+d)^(1/2)/(c*x^4+a)^(1/2)-1/8*(-2*B*e+C*d)*(a* 
e^2+c*d^2)^(1/4)*(a^(1/2)*e-(a*e^2+c*d^2)^(1/2))^2*(1+a^(1/2)*(e+d/x^2)/(a 
*e^2+c*d^2)^(1/2))*((c+a/x^4)/(c+a*e^2/d^2)/(1+a^(1/2)*(e+d/x^2)/(a*e^2+c* 
d^2)^(1/2))^2)^(1/2)*(e+d/x^2)^(1/2)*x^3*EllipticPi(sin(2*arctan(a^(1/4)*( 
e+d/x^2)^(1/2)/(a*e^2+c*d^2)^(1/4))),1/4*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^2 
/a^(1/2)/e/(a*e^2+c*d^2)^(1/2),1/2*(2+2*a^(1/2)/(a*e^2+c*d^2)^(1/2)*e)^...
 

Mathematica [F]

\[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \sqrt {a+c x^4}} \, dx=\int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \sqrt {a+c x^4}} \, dx \] Input:

Integrate[(A + B*x^2 + C*x^4)/(Sqrt[d + e*x^2]*Sqrt[a + c*x^4]),x]
 

Output:

Integrate[(A + B*x^2 + C*x^4)/(Sqrt[d + e*x^2]*Sqrt[a + c*x^4]), x]
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {A+B x^2+C x^4}{\sqrt {a+c x^4} \sqrt {d+e x^2}} \, dx\)

\(\Big \downarrow \) 2261

\(\displaystyle \int \frac {A+B x^2+C x^4}{\sqrt {a+c x^4} \sqrt {d+e x^2}}dx\)

Input:

Int[(A + B*x^2 + C*x^4)/(Sqrt[d + e*x^2]*Sqrt[a + c*x^4]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2261

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol 
] :> Unintegrable[Px*(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, 
p, q}, x] && PolyQ[Px, x]
 

Maple [F]

\[\int \frac {C \,x^{4}+B \,x^{2}+A}{\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+a}}d x\]

Input:

int((C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(c*x^4+a)^(1/2),x)
 

Output:

int((C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(c*x^4+a)^(1/2),x)
 

Fricas [F]

\[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \sqrt {a+c x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {c x^{4} + a} \sqrt {e x^{2} + d}} \,d x } \] Input:

integrate((C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(c*x^4+a)^(1/2),x, algorithm="fr 
icas")
 

Output:

integral((C*x^4 + B*x^2 + A)*sqrt(c*x^4 + a)*sqrt(e*x^2 + d)/(c*e*x^6 + c* 
d*x^4 + a*e*x^2 + a*d), x)
 

Sympy [F]

\[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \sqrt {a+c x^4}} \, dx=\int \frac {A + B x^{2} + C x^{4}}{\sqrt {a + c x^{4}} \sqrt {d + e x^{2}}}\, dx \] Input:

integrate((C*x**4+B*x**2+A)/(e*x**2+d)**(1/2)/(c*x**4+a)**(1/2),x)
 

Output:

Integral((A + B*x**2 + C*x**4)/(sqrt(a + c*x**4)*sqrt(d + e*x**2)), x)
 

Maxima [F]

\[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \sqrt {a+c x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {c x^{4} + a} \sqrt {e x^{2} + d}} \,d x } \] Input:

integrate((C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(c*x^4+a)^(1/2),x, algorithm="ma 
xima")
 

Output:

integrate((C*x^4 + B*x^2 + A)/(sqrt(c*x^4 + a)*sqrt(e*x^2 + d)), x)
 

Giac [F]

\[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \sqrt {a+c x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {c x^{4} + a} \sqrt {e x^{2} + d}} \,d x } \] Input:

integrate((C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(c*x^4+a)^(1/2),x, algorithm="gi 
ac")
 

Output:

integrate((C*x^4 + B*x^2 + A)/(sqrt(c*x^4 + a)*sqrt(e*x^2 + d)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \sqrt {a+c x^4}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{\sqrt {c\,x^4+a}\,\sqrt {e\,x^2+d}} \,d x \] Input:

int((A + B*x^2 + C*x^4)/((a + c*x^4)^(1/2)*(d + e*x^2)^(1/2)),x)
 

Output:

int((A + B*x^2 + C*x^4)/((a + c*x^4)^(1/2)*(d + e*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \sqrt {a+c x^4}} \, dx=\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+a}\, x^{4}}{c e \,x^{6}+c d \,x^{4}+a e \,x^{2}+a d}d x \right ) c +\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+a}\, x^{2}}{c e \,x^{6}+c d \,x^{4}+a e \,x^{2}+a d}d x \right ) b +\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+a}}{c e \,x^{6}+c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a \] Input:

int((C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(c*x^4+a)^(1/2),x)
 

Output:

int((sqrt(d + e*x**2)*sqrt(a + c*x**4)*x**4)/(a*d + a*e*x**2 + c*d*x**4 + 
c*e*x**6),x)*c + int((sqrt(d + e*x**2)*sqrt(a + c*x**4)*x**2)/(a*d + a*e*x 
**2 + c*d*x**4 + c*e*x**6),x)*b + int((sqrt(d + e*x**2)*sqrt(a + c*x**4))/ 
(a*d + a*e*x**2 + c*d*x**4 + c*e*x**6),x)*a